Quadratic autocatalysis fronts

We begin by considering the simplest form of autocatalysis, which is characterized by a rate equation with a quadratic nonlinearity  

In a distributed, unstirred solution, where reaction couples with diffusion, the space and time evolution of the system is described by the partial differential equations 

It is convenient to express the reaction-diffusion equation in dimensionless terms by scaling the concentration variables according to the initial reactant concentration ahead of the wave, a = [A]/ io, = [B]/ao, and the space and time according to = x kgaQland t = kgagt  

8 = DJD is the ratio of the diffusion coefficients, = d /d z is the one-dimensional Laplacian, and % is the initial concentration the reactant. These equations describe a front propagating in the positive 4 direction, where the boundary conditions are given by 

The net reaction of the autocatalytic process [76] is A B therefore, in the absence of diffusion or in the case of equal diffusivities, the reactant and product concentrations are linked by the conservation relation [A]q = [A] -f [B]. In terms of the dimensionless variables and the above initial conditions, we have the relation a + p = 1 for the case of equal diffusivities, 8 = 1. Hence, the 

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Figure 17 Phase plane for quadratic autocatalysis front described by Eqs. [83]. The front profile corresponds to a trajectory emanating from the origin (saddle point) along the outset and approadiing the singularity at (1, 0) along the (degenerate) eigenvector (Reprinted from Ref. 43 with permission of the American...

Hence for quadratic autocatalysis, constant velocity, constant wave-form propagating fronts are allowed with any velocity c greater than some minimum velocity ... 

The behavior found for the quadratic system in the one-dimensional case is directly applicable to the two-dimensional configuration planar fronts are exhibited with velocities given by Eq. [88] for the case of equal diffusivities of A and B. When the diffusivities significantly differ, planar fronts are still observed however, now the velocity scales with the diffusion coefficient D = Dr according to Eq. [89]. > The one-dimensional solution for the cubic system is also valid for the two-dimensional configuration however, the cubic front may exhibit lateral instabilities that are not observed in the quadratic system. will now consider the stability of cubic autocatalysis fronts. 

The front velocity for cubic autocatalysis may take on any value above some minimum velocity just as for the quadratic case. In dimensionless terms, we have... 

The simplest attack [7] anticipates the fact that cubic autocatalysis generates a steady wave-front of quadratic form ... 

Next, we consider the wavefronts that may develop for two other geometries a circle and a sphere. These represent the simplest 2-D and 3-D geometries. The natural response to circular or spherical initiations at the center of such reaction zones will be the development of fronts with the same underlying shape. We concentrate here on quadratic and cubic autocatalysis without decay. 

This diffusive instability mechanism has only recently been examined in reaction-diffusion systems. Continuing with the analogy between isothermal fronts and nonisothermal flames, we pursue the case where > 1 in Equations (4). Thus, the front is destabilized when the diffusivity of the reactant A becomes sufficiently larger than that of the autocatalyst B. We consider systems with pure cubic autocatalysis here in a detailed study [12], pure quadratic and mixed-order fronts are also considered and found to have different sensitivities to the destabilizing effects of diffusion. 

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